277 Symmetry
A mapping that leaves an equation unchanged that is used to simplify equations.
Symmetry is about invariance to coordinate systems.
definition (Symmetry) A property of a tensor or matrix where its components remain unchanged when their indices are interchanged or when the object is transposed, satisfying the following conditions:
- (Tensor Component Form) For a second-rank tensor \(A\), it is symmetric if \(A_{mn} = A_{nm}\) for all \(m\) and \(n\).
- (Matrix Transposition) A matrix is symmetric if it is unchanged by transposition, such that \(\tilde{A} = A\).
- (Resolution Property) Every second-rank tensor can be resolved into a symmetric part, given by \(\dfrac{1}{2}(A_{mn} + A_{nm})\), and an antisymmetric part.
where - \(A\) is a tensor. - \(m, n\) are indices identifying components. - \(\tilde{A}\) is the transpose of the matrix.
Note:
- \(A\) may be a matrix.
- \(\tilde{A}\) is also written \(A^T\).
definition (Antisymmetry) A property of a tensor or matrix where its components change sign when their indices are interchanged, satisfying the following conditions:
- (Tensor Component Form) For a second-rank tensor \(A\), it is antisymmetric if \(A_{mn} = -A_{nm}\) for all \(m\) and \(n\).
- (Diagonal Property) For any antisymmetric matrix, the diagonal elements (where \(m=n\)) must be zero.
- (Dual Association) In 3-D space, every antisymmetric second-rank tensor \(C\) can be associated with a dual pseudovector \(\mathbf{C}\).
where - \(A\) is a tensor. - \(m, n\) are indices identifying components. - \(C\) is an antisymmetric tensor.
Note:
- \(A\) may be a matrix.
definition (Permutational Antisymmetry = Fermion Antisymmetry) A property of identical-particle systems, specifically fermions, whose wave functions must change sign under pairwise particle interchanges, satisfying the following conditions:
- (Permutation Condition) A many-fermion wave function \(\Psi_F(1, \dots, n)\) must satisfy \(P\Psi_F(1, \dots, n) = \epsilon_P\Psi_F(1, \dots, n)\) for any permutation \(P\).
- (Representation Role) \(\Psi_F\) serves as the sole basis function for the totally antisymmetric representation of the symmetric group \(S_n\).
- (Levi-Civita Relation) The sign of the transformation is determined by the Levi-Civita symbol \(\epsilon_P\), where \(\epsilon_P = 1\) for even permutations and \(\epsilon_P = -1\) for odd permutations.
where - \(\Psi_F\) is a fermion wave function. - \(P\) is a permutation operator acting on particle numbers. - \(\epsilon_P\) is the parity of the permutation. - \(S_n\) is the symmetric group of order \(n!\).